Probability and Measure

By R. Keener. 2009

2,409 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Data Encoding for Byzantine-Resilient Distributed Optimization (2019, background): “It is well-known that a distribution is absolutely continuous if and only if it can be represented as an integral over an integrable density function [Bil95, Theorem 31.8, Chapter 6].”
A Distributional Perspective on Reinforcement Learning (2017, background): “The reader unfamiliar with mea-A sure theory may think of Ω as the space of all possible outcomes of an experiment (Billingsley, 1995).”
Differentially Private Empirical Risk Minimization (2009, neutral): “This ratio can be written as (Billingsley, 1995) g(fpriv”
Testing for Homogeneity with Kernel Fisher Discriminant Analysis (2007, background): “(54) Finally, combining (51) and (53) (54), by the triangular inequality, we have proved that, for ǫ > 0, we may choose a sufficiently large N , such that |E[eiuV̂n,∞(γ)] − E[eiuVn,∞(γ)]| ǫ , (55) and the proof is concluded by invoking Lévy’s continuity theorem (Billingsley, 1995, Theorem 26.3).”
Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization (2008, methodology): “We also use the well known Fatou’s lemma [38], stating that for a sequence of non-negative random variables {Xk}, we have”
How to Start Training: The Effect of Initialization and Architecture (2018, methodology): “We have where we can replace the sigma algebra F d − 1 by the sigma algebra generated by act ( d − 1) since the computation done by a feed-forward neural net is a Markov chain with respect to activations at consecutive layers (for background see Chapter 8 in [3]).”
Gaussian Process Behaviour in Wide Deep Neural Networks (2018, methodology): “The Cram´er-Wold device (Cram´er and Wold, 1936) (Billingsley, 1986, p. 383) states that convergence of a sequence of ﬁnite-dimensional vectors to some limit is equivalent to convergence on all possible linear projections to the corresponding real-valued random variable.”
Central Moment Discrepancy (CMD) for Domain-Invariant Representation Learning (2017, background): “Concerning approximation (c), the joint application of the weak law of large numbers (Billingsley, 2008) with the continuous mapping theorem (Billingsley, 2013) proves that this approximation creates a consistent estimate.”
Measure Transformer Semantics for Bayesian Machine Learning (2011, methodology): “To give a formal semantics to such observations, as well as to mixtures of continuous and discrete distributions, we turn to measure theory, following standard sources [3].”
Importance Nested Sampling and the MultiNest Algorithm (2013, background): “…in such single draw importance sampling from each of the speciﬁed h k,n ( · ) has ﬁnite (but non-zero) variance (cf. Hesterberg 1995), i.e., and that together our ˆ Z ( n ) k satisfy Lindeberg’s condition such that the CLT holds for this triangular array of random variables (cf. Billingsley 1995).”
Extended Conditional Independence and Applications in Causal Inference (2015, methodology): “We will show that D A X is a d -system that contains Π; it will then follow from Dynkin’s lemma [Billingsley, 1995, p. 42] that D A X contains σ (Π) = σ ( Y, Z ) and hence that (A.1) holds.”
Batch means and spectral variance estimators in Markov chain Monte Carlo (2008, positive): “Now a standard Borel-Cantelli argument (pp 59-60 Billingsley (1995)) yields  1 n bn−1 ∑”
Private Convex Optimization for Empirical Risk Minimization with Applications to High-dimensional Regression (2012, background): “Caratheodory's Extension Theorem (Billingsley, 1995) guarantees that µ(E) = µ * (E)”
On Sparse Variational Methods and the Kullback-Leibler Divergence between Stochastic Processes (2015, background): “Readers looking for some background on these issues may wish to consult a larger text (Billingsley, 1995; Capinski and Kopp, 2004).”